This website implements the sensitivity analyses described in
Mathur & VanderWeele (2020a),
Mathur & VanderWeele (2020b), and
VanderWeele & Ding (2017).
Sensitivity analysis for the pooled point estimateThis tab computes the E-value for the pooled point estimate of a meta-analysis (Section 7.2 of
Mathur & VanderWeele, 2020a; see
VanderWeele & Ding (2017) and
this page for more on E-values in general). This meta-analysis E-value represents the average severity of confounding in the meta-analyzed studies (i.e., the minimum strength of association
on the risk ratio scale that unmeasured confounder(s) would need to have with both the exposure
and the outcome, conditional on the measured covariates), to fully explain away the observed meta-analytic point estimate in the sense of shifting it to the null. Note that for outcome types other than relative risks, assumptions
are involved with the approximate conversions used. See
VanderWeele & Ding (2017) for details.
Alternatively, you can consider the average confounding strength required to reduce the observed
point estimate to any other value (e.g. attenuating the observed association to a true causal
effect that is no longer scientifically important, or alternatively increasing a near-null
observed association to a value that is of scientific importance). For this purpose, simply
type a non-null effect size into the box "True causal effect to which to shift estimate"
when computing the meta-analytic E-value.
Interpreting the resultsFor example, if your meta-analytic point estimate on the relative risk scale is 1.5 (95% confidence interval: [1.4, 1.5]), you will obtain an E-value for the point estimate of 2.37 and an E-value for the lower confidence interval limit of 2.15. This means that if, hypothetically, the meta-analyzed studies were subject to confounding such that, on average across the studies, there were unmeasured confounder(s) that were associated with the studies' exposures and outcomes by relative risks of at least 2.37 each, this amount of average confounding could potentially explain away the point estimate of 1.5 (i.e., to have the true causal effect be a relative risk of 1), but weaker average confounding could not. Similarly, if this strength of average confounding were at least 2.15 across studies, this amount of confounding could potentially shift the confidence interval to include the null, but weaker average confounding could not.
A caveat about the pooled point estimateNote that this tab of the website conducts sensitivity analyses that describe evidence strength only in terms of the pooled point estimate, a measure that does not fully characterize effect heterogeneity in a meta-analysis. For example, consider two meta-analyses with the pooled point estimate of relative risk = 1.1. The first, Meta-Analysis A, has very little heterogeneity, such that all true population effects are very close to 1.1. In contrast, despite having the same point estimate, Meta-Analysis B could have substantial heterogeneity, such that a large proportion of the true population effects are of scientifically meaningful size (e.g., >1.2). Thus, Meta-Analysis B provides stronger support for the presence of meaningfully strong effects than does Meta-Analysis A, and furthermore Meta-Analysis B might also suggest that a non-negligible proportion of the effects are actually preventive rather than causative (i.e., with relative risks less than 1). For this reason, meta-analyses that have some heterogeneity should generally report not only the point estimate, but also the estimated percentage of meaningfully strong population effects
(Mathur & VanderWeele, 2019)., and sensitivity analyses should consider this quantity as well (which you can do using the tab "Sensitivity analysis for the proportion of meaningfully strong effects").
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: You are calculating a "non-null" E-value, i.e., an E-value for the minimum
amount of unmeasured confounding needed to move the estimate and confidence interval
to your specified true value rather than to the null value.
Note: Using the standard deviation of the outcome yields a conservative approximation
of the standardized mean difference. For a non-conservative estimate, you could instead use the estimated residual standard deviation from your linear
regression model. Regardless, the reported E-value for the confidence interval treats the
standard deviation as known, not estimated.